Open Journal of Fluid Dynamics, 2013, 3, 42-47
http://dx.doi.org/10.4236/ojfd.2013.32A007 Published Online July 2013 (http://www.scirp.org/journal/ojfd)
Numerical Study on Transonic Flow with Local
Occurrence of Non-Equilibrium Condensation
Shigeru Matsuo1, Kazuyuki Yokoo2, Junj i Nagao2, Yushiro Nishiyama2,
Toshiaki Setoguchi3, Heuy Dong Kim4, Shen Yu5
1Department of Advanced Technology Fusion, Saga University, Saga, Japan
2Graduate School of Science & Engineering, Saga University, Saga, Japan
3Institute of Ocean Energy, Saga University, Saga, Japan
4School of Mechanical Engineering, Andong National University, Andong, Korea
5Institute of Engineering Thermophysics, Chinese Academy of Science, Beijing, China
Received May 24, 2013; revised June 1, 2013; accepted June 8, 2013
Copyright © 2013 Shigeru Matsuo et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Characteristics of transonic flow over an airfoil are determined by a shock wave standing on the suction surface. In this
case, the shock wave/boundary layer interaction becomes complex because an adverse pressure gradient is imposed by
the shock wave on the boundary layer. Several types of passive control techniques have been applied to shock
wave/boundary layer interaction in the transonic flow. Furthermore, possibilities for the control of flow fields due to
non-equilibrium condensation have been shown so far, and in this flow field, non-equilibrium condensation occurs
across the passage of the nozzle and it causes the total pressure loss in the flow field. However, local occurrence of
non-equilibrium condensation in the flow field may change the characteristics of total pressure loss compared with that
by non-equilibrium condensation across the passage of flow field and there are few for researches of locally occurred
non-equilibrium condensation in a transonic flow field. The purpose of this study is to clarify the effect of locally oc-
curred non-equilibrium condensation on the shock strength and total pressure loss on a transonic internal flow field with
circular bump. As a result, it was found that shock strength in case with local occurrence of non-equilibrium condensa-
tion is reduced compared with that of no condensation. Further, the amount of increase in the total pressure loss in case
with local occurrence of non-equilibrium condensation was also reduced compared with that by non-equilibrium con-
densation across the passage of flow field.
Keywords: Compressible; Transonic; Shock Wave; Non-Equilibrium Condensation; Simulation
1. Introduction
A shock wave standing on the suction surface of airfoils
determines the characteristics of transonic flow field. In
this case, the shock wave imposes an adverse pressure
gradient on the boundary layer and it makes the shock
wave/boundary layer interaction complex. Several types
of passive control techniques have been proposed to con-
trol the shock wave/boundary layer interaction in the
transonic flow field. For instance, Bahi et al. [1] and
Raghunathan [2] described that a porous wall and cavity
system that applied at the foot of the shock wave was
effective in decreasing undesirable adverse pressure gra-
dient of the shock wave/boundary layer interaction.
However, this control method essentially leads to large
viscous losses caused by the porous walls. These losses
may be greater than the control benefit of the shock wave.
Thus, the method cannot be generalized as an effective
method. In order to overcome the above demerits, Saida
et al. [3] proposed several techniques and showed that
the porous wall with a cavity and vortex generator to the
shock wave/boundary layer interaction was effective
method to reduce the wave drag and suppress the devel-
opment of the boundary layer. Further, Raghunathan [4]
and O'Rourke et al. [5] reported that the passive control
using the porous wall with a cavity and vortex control
jets upstream of porous wall might be effective control
method for the shock position and pressure gradient.
Furthermore, the control method using non-equilib-
rium condensation has been proposed by several re-
C
opyright © 2013 SciRes. OJFD

S. MATSUO ET AL. 43
searches (Wegener et al. [6], Matsuo et al. [7], Sislian [8],
Setoguchi et al. [9], Matsuo et al. [10], Tanaka et al.
[11]). As is evident from these researches, it is known
that non-equilibrium condensation can reduce the de-
velopment of boundary layer behind the shock wave. In
the flow fields, non-equilibrium condensation occurs
across the passage of the nozzle and it causes the total
pressure loss in the flow field. However, local occurrence
of non-equilibrium condensation in the flow field may
change the characteristics of total pressure loss compared
with that by non-equilibrium condensation across the
passage of flow field. However, there are few for re-
searches of locally occurred non-equilibrium condensa-
tion in a transonic flow field.
The objectives in the present study is to clarify the ef-
fect of locally occurred non-equilibrium condensation on
the shock strength and total pressure loss in a transonic
internal flow field with a circular bump.
2. CFD Analysis
Governing Equations
Following assumptions were used in the present study;
both velocity slip and temperature difference do not exist
between condensate particles and gas mixture, and the
effect of the condensate particles on pressure is neglected.
The governing equation, i.e., the unsteady 2D com-
pressible Navier-Stokes equations that were combined
with nucleation rate, a droplet growth and diffusion equa-
tions (Bird et al. [12], Hirschfelder [13]) written in
dd
1dd d
S
t
S
Re

 



 



QH
RI S
(1)
where
1
1k
J
u
g
n






































Q
t
v
E
,

1
1
x
y
t
uU n p
vU n
EpU
kU
JU
gU
nU
U























H
p
U 


,
*
11
11 1
0
0
0
xx xxyy
yx xyyy
xx yy
lx
lx
xy
nn
nn
fn fn
kk k
nn
xy
knn
xy
cc
nn
xy




 



y
y
























































 












R
*
2
d
0
0
0
0
1
1
1
0
0
0
ke
e
ke e
R


e
PkR
R
Jkk
PR
k Rxxyy
















 

 


 







I,
32
0
0
0
0
0
1
0
4d
π3
3d
0
lcF
F
J
r
rInr t
I





































S (2)
In Equation (1), Q is conservative vector, H is inviscid
flux vector and R is viscous flux vectors. I and S are the
source terms corresponding to turbulence and condensa-
tion, respectively. τxx, τxy, τyx and τyy are components of
viscous shear stress.
In Equation (2),

S. MATSUO ET AL. 44



j
x
xxxx xy yyyx yy
fnqu vnquv

  
(3)
1
111 1cx y
cc
nn
1
x
y
 











(4)
t12
112 2
tjj j
1
1
j
x
cc T
qh
Pr Prxxx




 








h




(5)
*
k
k
f



 , k
f




 (6)
2
Hdk
kk
fP
kxxyy








 




(7)
2
2
t
1
22
k
uvuv
Pxxyy





 



 






(8)
t
k



, i, ji, j
2
max *
SS








(9)
k
i, ji, j
k
1
3
u
SS
x


, j
i
i, j
ji
1
2
u
u
S
x
x










2
(10)
The density of gas mixture is calculated by the sum of
density of vapor (ρ1) and dry air (ρ2);
1

  (11)
The mass fraction can be given as
i
i
c


 (12)
In Equations (4)-(6), Δ1 and Δ2 are effective diffusivi-
ties of vapor and dry air, respectively. The closure coef-
ficients are,
t
91
0.0708,*,,
100 2
37
*,
58
Pr





(13)
jj
d
d0 d0
jj
0, 0
1
,08
k
xx
k
xx







 


 

(14)
The governing equation system are non-dimension-
alized with reference values at the reservoir condition.
Implicit upwind relaxation scheme (Furukawa et al. [14])
is used to solve the governing equations. The equations
were discretized in time using the Euler implicit method
and in space using a cell-centered finite volume method
with a quadrilateral structured cell system. A third-order
TVD scheme with MUSCL (Yee [15]) was used to dis-
cretize the spatial derivatives, and second-order central
difference scheme for the viscous term. In the solver, the
relaxation was performed with a point Gauss-Seidel te-
chnique. To close the governing equations, k-ω model
(Wilcox [16]) was employed in computations.
3. Computational Conditions
Figure 1 shows a computational domain of the transonic
flow field and boundary condition. The test section has a
height of H = 60 mm at the inlet and exit. The radius of
circular arc of the nozzle is R = 100 mm and the height of
nozzle throat H* is 56 mm. The region upstream of the
nozzle was separated into dry air and moist air regions by
plate. Thickness of the plate is 0 mm. Furthermore,
working gas (dry air, moist air) in a tank flows into the
main flow from the leading edge of the circular bump
through the narrow passage (d = 3 mm)
Table 1 shows initial conditions used in the present
calculation. Total pressure p0 and temperature T0 at stag-
nation point upstream of the nozzle are 102 kPa and 287
K, respectively.
The working gases of upper and lower sides of the
plate are dry and moist airs, respectively. Values of Y/H
Figure 1. Computational domain and boundary condition.
Table 1. Initial conditions.
T0 = 287 K
Case No. p0t/p0d [mm] Y/H Initial degree
of supersaturation
Case 1-D 0 S0u = S0l = 0
Case 1-M-A1
Case 1-M-B0.0625
Case 1-M-C
- 0
0.0313
S0u = 0 S0l = 0.8
Case 2-D S0t = 0
Case 2-M3
0.813 0
S0t = 0.8

S. MATSUO ET AL. 45
which shows the plate position, are 0 (Case 1-D, 2-D3,
2-M3), 1.0 (Case 1-M-A), 0.0625 (Case 1-M-B) and
0.0313 (Case 1-M-C). Furthermore, stagnation pressure
p0t in the rank is 82.62 kPa and working gas in the tank is
dry air (Case 2-D3) or moist air (Case 2-M3). Initial de-
gree of supersaturation (S0, S0t) of moist air is 0.8.
The number of grids is 450 × 228. The adiabatic no-
slip wall was used as boundary condition. The boundary
conditions of inlet and exit were fixed at initial condition
and out flow condition, respectively. Condensate mass
fraction g was set at g = 0 on the wall.
4. Result and Discussion
Figure 2 shows schlieren photographs obtained by ex-
periment (Figures 2(a) and (c)) and computer schlieren
pictures (Figures 2(b) and (d)). In Figures 2(a) and (b),
working gas is dry air (Experiment: S0 = 0.18, Simula-
tion: S0 = 0) and moist air (S0 = 0.5) in Figures 2(c) and
(d). Flow direction is left to right. As seen from these
figures, shock wave is observed on the circular arc bump.
In the case of dry air (Figures 2(a) and (b)), the shock
wave is clearly visible compared with that of moist air
(Figures 2(c ) and (d)).
Figures 3(a) and (b) show static pressure distributions
on the lower wall in cases of dry air and moist air, re-
spectively. In each figure, comparison between the ex-
perimental and simulated static pressure distributions are
shown and it is found from these figures that simulated
results agree well with experimental results.
Figure 4 shows time-averaged contour maps of Mach
number (line) and condensate mass fraction g (gray) for
all cases. As seen from Figures 4(b)-(d) and (f), con-
densate mass fraction (condensate droplet) begins to in-
Flow
(a) (c)
Flow
(b) (d)
Flow Flow
Figure 2. Comparison between the experiment and simu-
lated results. (a) Experiment (S0 = 0.18); (b) Simulation (S0
= 0.0); (c) Experiment (S0 = 0.5); (d) Simulation (S0 = 0.5).
(a)
(b)
Figure 3. Comparison between the experimental and simu-
lated static pressure distributions on the lower wall. (a) Dry
air; (b) Moist air.
x/H*
y/H*
(a)
x/H*
y/H*
(b)
x/H*
y/H*
(c)
x/H*
y/H*
(d)
x/H*
y/H*
(e)
x/H*
y/H*
(f)
-10 1
0
0.5
1
-1 01
0
0.5
1
-10 1
0
0.5
1
-1 01
0
0.5
1
-1 01
0
0.5
1
-1 01
0
0.5
1
Figure 4. Contour maps of Mach number (line) and con-
densate mass fraction (Grey). (a) Case 1-D; (b) Case 1-M-A;
(c) Case 1-M-B; (d) Case 1-M-C; (e) Case 2-D3; (f) Case
2-M3.

S. MATSUO ET AL. 46
crease along the bump wall side upstream of the shock
wave and distributes over downstream region, and it ex-
pands in the positive direction of y-axis in order of Cases
1-M-C, 1-M-B and 1-M-A. Furthermore, for Cases 1-M-
A (Figure 4(b)) to 2-M3 (Figure 4(f)), the height of adi-
abatic shock wave seems to become small compare with
that for Case 1-D (Figure 4(a )).
Figure 5 shows distributions of shock strength (
 =
p2/p1) in the direction of y-axis for all cases. As seen
from this figure, shock strength for Case 1-D is the larg-
est in all cases and it decreases in order of Case 1-M-C,
1-M-B and 1-M-A. However, there is no difference of
the strength between Cases 1-M-C and 1-M-B, as well as
the difference between Cases 2-D3 and 2-M3. It is found
from this result that the strength is changed by occur-
rence region of non-equilibrium condensation.
Figure 6 shows distributions of integrated total pres-
sure loss β in the direction of x-axis. Integrated total pres-
sure loss is calculated from following equation:
Upper wall01
Lower wall0
1- d
py
p




 (15)
where p01 and p0 indicate local and stagnation total pres-
sures, respectively.
From this figure, integrated total pressure losses of
Cases 1-M-A, 1-M-B and 1-M-C begin to deviate from
the distribution of Case 1-D at the position just upstream
of the shock wave. It is considered that this is due to an
increase of entropy by non-equilibrium condensation
occurred upstream of the shock wave. Integrated total
pressure losses for Cases 1-M-C is small compared with
that of 2-D3 and 2-M3. Further, integrated total pressure
losses of Cases 2-D3 and 2-M3 begin to deviate from the
distribution of Case 1-D at the position close to the lead-
ing edge of the circular bump. This is due to effect of
Figure 5. Distributions of shock strength  .
blowing of the flow through the narrow passage. Based
on the results in Figures 5 and 6, it is considered from
the point of view of energy loss that the flow of Case
1-M-C is the most effective in all cases.
Figure 7 shows time-averaged distributions of dis-
placement thickness δ*/H* for all cases. The abscissa is
the distance x/H* from throat, and the ordinate is dis-
placement thickness δ*/H*. As is evident from this fig-
ure, displacement thickness behind the shock wave for
Case 1-M-A is the smallest in all cases and in Cases
2-D3 and 2-M3, it is high compared with that for other
cases in −0.5 ≤ x/H* ≤ 0.5. From these results, it is found
that the development of boundary layer is dependent on
how the non-equilibrium condensation occurs.
5. Conclusions
A numerical study has been made to investigate the ef-
fect of locally occurred non-equilibrium condensation on
a transonic flow field with a circular arc bump. The
10.50-0.5-1
x
/
H
*
0
0.01
0.02
0.03
0.05
0.04
ß
: Case 1-D
: Case 1-M-A
: Case 1-M-B
: Case 1-M-C
: Case 2-D3
: Case 2-M3
: Shock wave
Figure 6. Distributions of integrated total pressure loss  .
: Case 1-D
: Case 1-M-A
: Case 1-M-B
: Case 1-M-C
: Case 2-D3
: Case 2-M3
: Shock wave
-0.5 -0.2500.250.5
0
0.0 2
0.0 4
0.0 6
0.0 8
x
/
H
*
d*/H*
Figure 7. Time-averaged distributions of displacement thick-
ness δ*/H*.

S. MATSUO ET AL. 47
results obtained are summarized as follows:
1) Locally occurred non-equilibrium condensation on
the transonic bump model weakened the shock strength.
2) Compared with the case of occurrence of non-equi-
librium condensation across the passage of the nozzle,
locally occurred non-equilibrium condensation could re-
duce the total pressure loss downstream of the shock
wave.
3) The development of boundary layer was dependent
on how the non-equilibrium condensation occurs.
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